Journal of Physics Condensed Matter

Three, four and five interacting electrons in a spherical box: an exact diagonalization study extended

David C. Thompson, Ali Alavi

Published Oct 30, 2004• 9 citations

Abstract

We study the problem of N electrons confined to a hard three-dimensional spherical box, for N = 3,4 and 5. Full-configuration interaction and Hartree–Fock calculations have been performed. We report energies, densities, second-order density matrices and exchange–correlation holes. In the N = 4 system, we observe a spin transition as the radius R of the sphere is increased, changing the ground state from 3 P to 5 S. The N = 3 and 5 systems remain 2 P and 4 S, respectively, for all R investigated. Pair-correlation functions and physical exchange–correlation holes have been computed over a regime in which the systems transform from liquid droplets to Wigner molecules. These functions show marked, yet characteristic, changes in this regime. By computing | E c / E |, where E c is the correlation energy defined with respect to the unrestricted Hartree–Fock solution, we show that the maximum relative error in the UHF energies occurs in this crossover regime. These systems provide challenging benchmarks for post-Hartree–Fock and density functional theory methods.

DOI: 10.1088/0953-8984/16/45/020

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Journal of Physics Condensed Matter

Three, four and five interacting electrons in a spherical box: an exact diagonalization study extended

David C. Thompson, Ali Alavi

Published Oct 30, 2004• 9 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/16/45/020

Related papers

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Physical Review AJul 22, 20037 citations

We study a model of N electrons confined to a hard spherical box of radius R, for

N=3, 4,

and 5. Unrestricted Hartree--Fock and full configuration-interaction calculations have been performed for high, medium, and low density regimes. Remarkable changes in electronic structure can be induced by varying the effective Coulomb coupling. A low-spin$\ensuremath{\rightarrow}$high-spin transition occurs in the case of

N=4

as R is increased. We demonstrate a close connection between the propensity to form hybrid orbitals and the onset of Wigner molecule formation.

k)

arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially

{r}_{s},

the Wigner--Seitz density parameter) for cases where there is no background potential

(k=0)

and for when $k\ensuremath{\ne}0.$ With

k=0

and small

{r}_{s},

we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of

{r}_{s}

and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of

{r}_{s}

and k.

1/{r}_{12}

operator in the form of a double Fourier-Bessel-Fourier-cosine series and using enough terms to achieve a relative accuracy of ${10}^{\ensuremath{-}10}$ for the eigenvalues. Energy-level ordering and electron densities of selected electronic states are reported for cylindrical cavities of varying size with a fixed 1:1 length-to-diameter ratio. A remarkable feature of this system is that convergence of the full CI energy with respect to the one-electron basis set varies dramatically with the extent of confinement: For small cavities, the rate of approaching the basis-set limit is similar to that for two-electron atoms but is significantly faster for large cavities. Formation of singlet and triplet Wigner molecules is observed in cylindrical cavities with radii greater than about 5 bohrs.

A system of

N

antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.